asc report no. 32/2009

ASC Report No. 32/2009

Diffusive and Nondiffusive Population Models

Ansgar Jungel

Institute for Analysis and Scientific Computing

Vienna University of Technology — TU Wien

www.asc.tuwien.ac.at ISBN 978-3-902627-02-5

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ASCTU WIEN

Diffusive and nondiffusive population models

Ansgar Jungel1

Institute for Analysis and Scientific Computing, Vienna University of Technology,Wiedner Hauptstr. 8-10, 1040 Wien, Austria; e-mail: [email protected]

1 Introduction

The modeling of populations is of great importance in ecology and economy,for instance, to describe predator-prey and competition interactions, to pre-dict the dynamics of cell divisions and infectious diseases, and to managerenewable resources (harvesting). Population models describe the change ofthe number of species due to birth, death, and movement from position toposition (in space) or from stage to stage (age, size, etc.). In this short sur-vey, we review some mathematical results for deterministic and continuouspopulation models. We concentrate on the following model classes:

• spatially homogeneous population models;• spatially inhomogeneous population models;• age- and size-structured population models; and• time-delayed population models.

The evolution of spatially homogeneous populations may be modeled byordinary differential equations, for instance, by the logistic-growth model. Theinteraction of competing populations can be described by a system of coupledequations, one of which is the famous Lotka-Volterra system, introduced inSection 2. Important questions, beside the wellposedness of the correspond-ing problems, include the stability of steady states and the biological conse-quences.

In a spatially heterogeneous setting, the population number varies in spaceand may diffuse in the environment. This gives the class of reaction-diffusionequations and their systems. Turing found that the stationary solution to adiffusion system may become unstable even if the steady state of the cor-responding system without diffusion is stable. Thus, the stability analysis ismuch more involved than in the spatially homogeneous case. Roughly speak-ing, in the long-time limit, one may have extinction or coexistence of thespecies (see Section 3 for details).

2 Ansgar Jungel

The solutions of the diffusive Lotka-Volterra competition model do notshow pattern formation. Hence, this model is not able to describe segregationphenomena. In Section 4, we review several cross-diffusion models which allowfor inhomogeneous steady states. Roughly speaking, such stationary solutionsexist if cross-diffusion is large compared to diffusion. The existence analysis ofcross-diffusion systems is complicated due to the strong nonlinear coupling andsince the diffusion matrices may be neither symmetric nor positive definite.Recently, some analytical tools have been developed to prove the existence ofglobal-in-time weak solutions. These tools are explained in detail since theyreveal interesting connections between the symmetrization of the diffusionmatrix and the existence of an entropy (Lyapunov functional). Section 4 isthe key section of this survey.

When the individuals of a population are not identical but can be distin-guished by their age, size, etc., we need to introduce structured populationmodels. In Section 5, we introduce some age-structured and size-structuredbalance equations of hyperbolic type. Following [121], results on the existenceand the long-time behavior of the solutions are presented.

The change of a population may be delayed due to maturation or regen-eration time, for instance. In Section 6 we consider time-delayed populationmodels, following partially [56]. If the time delay is not discrete but distributed,we arrive to nonlocal equations in which the term with the retarded variableis replaced by a convolution in time.

The field of population modeling has become so large that this surveycan review only a small part of the published modeling and mathematicaltopics. Many model classes and important issues will not be discussed. Forinstance, we ignore difference and matrix equations, stochastic approaches,and models including mutations, maturation structures, metapopulations, anddemographic or biomedical applications. For details about these topics, werefer to the monographs [15, 38, 73, 100, 113, 114, 117, 121].

2 Initial-value population models

First, we consider the population dynamics of a single species without inter-actions in a homogeneous environment. Let u(t) be the population number attime t ≥ 0. Its rate of change is given by the difference of the birth and deathrates. Assuming that these rates are proportional to the population number,we obtain the differential equation du/dt = au, first suggested by Malthus[101] in the 18th century, where a, which is typically a positive number, isthe effective growth rate of the population. Its solution models unlimited ex-ponential growth. The capacity of the environment (limited food supply orother environmental resources) can be taken into account by introducing aself-limiting term:

du

dt= u(a − bu), t > 0, u(0) = u0, (1)

Diffusive and nondiffusive population models 3

where b > 0 is a measure of the environment capacity. This equation, proposedby Verhulst [143] in the 19th century, is called the logistic-growth model. Itssolution u(t) converges to the so-called carrying capacity limit a/b as t → ∞(if u(0) > 0), which is a stable steady state.

Next, let us consider the population of two species u(t) and v(t). When theyare not interacting, its evolution is described in the logistic-growth approachby the equations

du

dt= u(a1 − b1u),

dv

dt= v(a2 − c2v), t > 0.

The coefficients b1 ≥ 0 and c2 ≥ 0 are called the intra-specific competitionconstants. In this situation, the populations are evolving independently fromeach other. When the species are interacting, we add competition terms pro-portional to the population numbers:

du

dt= u(a1 − b1u − c1v),

dv

dt= v(a2 − b2u − c2v), t > 0, (2)

where the new coefficients c1 and b2 model inter-specific competition or ben-efit, depending on their sign. We distinguish three types of interactions:

• Predator-prey model: c1 > 0, b2 < 0. This choice decreases the (effective)growth rate of the species u and increases the growth rate for v. Since thegrowth rate for v becomes larger when the population number u is large,v represents a predator species and u the prey. Clearly, taking c1 < 0 andb2 > 0 changes the roles of prey and predator.

• Competition model: c1 > 0, b2 > 0. The interaction terms are nonpositive,thus decreasing the growth rates of the species. This means that bothspecies are competing for the (food or environmental) resources.

• Mutualistic (or symbiotic) model: c1 < 0, b2 < 0. When the two speciesbenefit from the interactions, the growth rates are enhanced. In this sit-uation, the interaction terms are taken nonnegative and the constants c1

and b2 are negative.

A particular predator-prey model is obtained when the intra-specific com-petition vanishes, b1 = c2 = 0:

du

dt= u(a1 − c1v),

dv

dt= −v(α − βu), t > 0,

where we have set α := −a2 > 0 and β := −b2 > 0. The sign assumptionon a2 means that the predators will become extinct in the absence of theprey, since in this situation, dv/dt = −αv. This system is usually called theLotka-Volterra model, proposed by Volterra [144] and independently by Lotkato describe a chemical reaction [94] in the first half of the 20th century. It hasthe remarkable property of possessing a first integral,

βu + c1v − log(uαva1) = const.,

4 Ansgar Jungel

showing that the system admits positive oscillating solutions if u(0) > 0,v(0) > 0 [114, Chap. 3.1]. The predator-prey model with limited growth (i.e.(2) with a2 < 0 and b2 > 0) allows for two scenarios. Depending on the choiceof the parameters and the initial data, the predator population becomes ex-tinct and the prey population approaches its carrying capacity limit, or thereis coexistence of both species, i.e., (u(t), v(t)) converges to the asymptoticallystable steady state

s∗ =(a1c2 − a2c1

b1c2 − b2c1,b1a2 − b2a1

b1c2 − b2c1

)

(3)

as t → ∞. Notice that our sign assumption on b2 implies that b1c2 − b2c1 > 0.Similar results are valid for the competition model [145, Chap. 2].

In the mutualistic model, one distinguishes between the weak mutualis-tic case b1c2 > b2c1, in which the self-limitation, expressed by b1 and c2,dominates the mutualistic interaction, expressed by b2 and c1, and the strongmutualistic case, in which mutualism dominates self-limitation. In the formercase, there is generally stable coexistence, whereas in the latter case, thereare three scenarios: either at least one of the species become extinct, or thesolution (u(t), v(t)) converges to the steady state (3) as t → ∞, or (u(t), v(t))blows up in finite time [96].

The dynamics of Lotka-Volterra systems with more than two species ismore involved. For instance, chaos has been observed in models with fourcompeting species (see [142] and the references therein).

3 Reaction-diffusion population models

In the previous section, we have considered populations which are spatiallyhomogeneous. However, in a spatially heterogeneous environment, the popu-lation density will depend on space. Assuming that populations tend to moveto regions with smaller number density, it is reasonable to include diffusiveterms in the evolution equations, which may be justified as limiting expres-sions of a Brownian motion [117]. Then a single-species population densityu(x, t) evolves in the bounded domain Ω ⊂ R

n according to

ut − d∆u = uf(x, u), x ∈ Ω, t > 0, u(·, 0) = u0, (4)

supplemented with some boundary conditions, where ut abbreviates the timederivative ∂u/∂t. Often, homogeneous Neumann boundary conditions ∇u ·ν = 0 on ∂Ω are taken, where ν is the exterior unit normal on ∂Ω. Theseconditions signify that the number of individuals is fixed in the domain (nomigration occurs). Also homogeneous Dirichlet boundary conditions u = 0 on∂Ω can be used, expressing a very hostile environment at the boundary. Thecoefficient d > 0 is the diffusion constant, and f(x, u) is the growth rate percapita, depending on the population and the heterogeneous environment. A


typical example is the logistic-growth function f(x, u) = a(x)−b(x)u; then (4)is called the Fisher-Kolmogorov-Petrovsky-Piskunov equation, introduced byFisher [44] and studied by Kolmogorov et al. [78]. Reaction-diffusion modelsof type (4) have been also considered in physics, chemistry, ecology etc.; seethe monographs of Okubo and Levin [117] and Murray [113].

The function f(x, u) = a(x)− b(x)u is decreasing in u (if b(x) > 0). Somepopulation ecologists argue that the growth rate f may not be decreasing in ufor all u ≥ 0, but it may achieve a maximum at an intermediate density. Thisso-called Allee effect [3] may be caused by, for instance, shortage of mates atlow density, lack of effective pollination, or predator saturation [131]. Whereasin the logistic growth case (with a(x) > 0) there exists a unique nonnegativesteady state (positive for slow diffusion and zero for fast diffusion; see [19]),there may be two steady states when an Allee effect is present [131]. Matano[102] and Casten and Holland [20] showed that any stable steady state to (4)(with homogeneous Neumann boundary conditions) is constant if the domainΩ is convex.

The situation becomes much more complex when we consider systems ofequations,

ut − ∆(Du) = g(x, u), x ∈ Ω, t > 0, u(·, 0) = u0, (5)

where u ∈ Rm is a vector-valued function, D = diag(d1, . . . , dm) is a diagonal

matrix with constant coefficients di, and g = (gi) : Ω × Rm → R

m (m > 1).Turing found in his seminal work [141] that different diffusion rates di in aparabolic system, modeling the interaction of two chemical substances, maylead to inhomogeneous distributions of the reactants, which allows one tomodel a pattern structure. Moreover, even if the steady state of the differ-ential equation without diffusion is stable, the corresponding steady state ofthe diffusion system may become unstable and bifurcations may occur. Thisphenomenon is generally called diffusion-driven instability.

Similar to the scalar case, Kishimoto and Weinberger [76] proved that,in a convex domain, the system (5) with homogeneous Neumann boundaryconditions has no stable nonconstant steady state if ∂gi/∂uj > 0 for alli 6= j (cooperation-diffusion system); the same conclusion holds for m = 2if ∂gi/∂uj < 0 for i 6= j (competition-diffusion system). On the other hand,in nonconvex domains, stable nonconstant steady states may exist, see theworks [66, 67, 103] for certain dumbbell-shaped domains. For more references,we refer to [40].

One may ask if a diffusion-driven instability also occurs in Lotka-Volterradiffusion systems. For this, following Lou and Ni [97], we discuss the compe-tition model

ut − d1∆u = u(a1 − b1u − c1v), vt − d2∆v = v(a2 − b2u − c2v), (6)

for x ∈ Ω and t > 0, with initial and homogeneous Neumann boundaryconditions. The coefficients ai, bi, ci, di (i = 1, 2) are positive. This model

6 Ansgar Jungel

may be supplemented by adding a given common environmental potential φ,modeling territories in which the environmental conditions are more or lessfavorable. In this situation, the equation for u has to be replaced by

ut − div(d1∇u − e1u∇φ) = u(a1 − b1u − c1v),

and similar for the equation for v.It is known that the initial-boundary value problem from (6) has a unique

nonnegative smooth solution, see, e.g., [149] for systems with m equationsand general semilinearities. The long-time behavior depends on the valuesof the reaction coefficients, and there are, in contrast to the Lotka-Volterradifferential equations (2), three situations. Set A = a1/a2, B = b1/b2, C =c1/c2. Then [97]

• Extinction: A > maxB,C or A < minB,C. The solution (u(t), v(t))converges to (a1/b1, 0) or (0, a2/c2), respectively, uniformly as t → ∞.Thus, one species dominates and the other species becomes extinct.

• Weak competition: B > A > C. The solution (u(t), v(t)) converges to thesteady state s∗, defined in (3), uniformly as t → ∞. This means that bothspecies coexist.

• Strong competition: B < A < C. The steady states (a1/b1, 0) and (0, a2/c2)are locally stable, but s∗ is unstable. If the domain is convex, no stablepositive steady state exists.

In particular, in the weak competition case, the steady state s∗ is globallyasymptotically stable regardless of the values of the diffusion coefficients d1

and d2. In fact, there exists a Lyapunov functional which allows for a long-timeasymptotic analysis [89, 126]. Therefore, no nonconstant steady state existsfor any d1 and d2, and there is no pattern structure. For the Volterra modelwith diffusion, for any number of interacting populations, Murray [112] hasshown that the effect of uniform diffusion is to damp all spatial variations.General reaction rates and the stability of constant steady states have beenconsidered by Conway and Smoller [27]. The situation changes when cross-diffusion terms are present in (6), modeling the population pressures createdby the competitors; see Section 4 for details.

The asymptotic behavior of solutions to reaction-diffusion systems similarto (6) has been studied in several papers. For instance, the existence of travel-ing wave solutions in one-dimensional diffusive Lotka-Volterra predator-preymodels with a logistic growth condition was shown by Dunbar [37]. More re-cently, traveling waves for a reaction-diffusion system with one diffusion termomitted were analyzed by Ai and Huang [1].

The following diffusive mutualistic model was considered by Lou, Nagilaki,and Ni [96]:

ut − d1∆u = u(a1 − b1u + c1v), vt − d2∆v = v(a2 + b2u − c2v)

in Ω×(0,∞) with initial and homogeneous Neumann boundary conditions andai, bi, ci > 0. Compared to (6), the signs of the interaction terms are reversed,


expressing mutualistic interactions. Lou, Nagilaki, and Ni prove the interestingresult that, in the strong mutualistic case (see Section 2), the population ofthe species may blow up in finite time, although one or both species withexactly the same initial data would die out if no diffusion effects are takeninto account. The mathematical reason is that diffusion first averages u andv, possibly increasing the densities, and, after some time, the reaction termsdominate and may force the solution to blow up. As the diffusion initiates theblowup process at the first stage, this phenomenon is called diffusion-inducedblowup.

A related effect is diffusion-induced extinction. Iida et al. [65] have stud-ied the diffusive Lotka-Volterra model in the strong competition case. In theabsence of diffusion, if one species is initially superior to the other one, thesuperior species wipes out the other species. On the other hand, allowing fordiffusion (with the same diffusion rates), Iida et al. proved that the superiorspecies may become extinct. If the diffusion rates are different, the situationis more complicated, and we refer to [116] for details.

Finally, we remark that systems with more than two equations havebeen considered too. For instance, the coexistence of competing species ina reaction-diffusion system with one predator and two competing prey isanalyzed in [70], and the existence and stability of stationary and periodicsolutions to a reaction-diffusion system consisting of m species is proved in[39].

4 Cross-diffusion population models

The diffusive Lotka-Volterra competition model has no nonconstant steadystate for all possible diffusion rates, thus excluding biologically reasonablepattern structures. Inhomogeneous steady states may be obtained by takinginto account cross-diffusion terms instead of just adding pure diffusion tothe population models. In this section, cross-diffusion models are analyzed indetail.

First, let us consider cross-diffusion systems with constant diffusion rates,

ut − (d1∆u + d2∆v) = uf(x, u, v), vt − (d3∆u + d4∆v) = vg(x, u, v)

in Ω × (0,∞) with initial and homogeneous Neumann boundary conditions.Clearly, a necessary condition to have – at least local – existence of solutionsis that the diffusion matrix

(

d1 d2

d3 d4

)

is positive definite (see, e.g., [7]). The above system shows indeed diffusion-driven instabilities. Farkas [41] has proved that the one-dimensional stationarymodel undergoes a Turing bifurcation at a certain size of the interval undersuitable conditions on the coefficients of the system. This means that a larger

8 Ansgar Jungel

domain may lead to a heterogeneous distribution of the steady states evenif the conditions are homogeneous everywhere. We also refer to [62, 80] forresults in this direction. The stability and cross-diffusion-driven instability ofconstant stationary solutions was studied by Flavin and Rionero [45]. Alsowavelike solutions are possible in such cross-diffusion systems. For instance,Kopell and Howard [79] have proved the existence of plane-wave solutions ofthe type (u, v)(x, t) = (U, V )(k · x−ωt), and they have analyzed the stabilityand instability of the waves. Summarizing, we may say that diffusion tendsto suppress pattern formation, whereas cross-diffusion seems to help creatingpattern under suitable conditions.

Generally, we expect that the cross-diffusion rate of one species is notconstant but depends on the population density of the other species and viceversa. Therefore, we replace the linear term by a nonlinear one involving theproduct of both populations. This leads to the following system, first suggestedby Shigesada, Kawasaki, and Teramoto [132]:

ut − ∆(

(d1 + α11u + α12v)u)

= u(a1 − b1u − c1v), (7)

vt − ∆(

(d2 + α21u + α22v)v)

= v(a2 − b2u − c2v), (8)

∇u · ν = ∇v · ν = 0 on ∂Ω, t > 0, u(·, t) = u0, v(·, t) = v0 in Ω, (9)

where Ω ⊂ Rn is a bounded domain. The diffusion coefficients di and αij

as well as the reaction coefficients ai, bi, and ci are assumed to be constant(and nonnegative). The expressions α12∆(uv) and α21∆(uv) are the nonlinearcross-diffusion terms, and α11∆(u2) and α22∆(v2) describe the self-diffusion ofthe species. The basic idea is that the primary cause of dispersal is migrationto avoid crowding instead of just random motion (modeled by diffusion). Inthe following, we review some mathematical properties of the above cross-diffusion system.

Stability. Mathematicians started to pay attention to the model (7)-(8) fromthe 1980s on, first examining mainly stability issues. One of the first pa-pers is due to Mimura and Kawasaki [109], who have shown, neglecting self-diffusion and assuming reaction coefficients such that b1 = b2, c1 = c2, andc1/b1 > a1/a2 > b1/c1, that the stationary one-dimensional system has spa-tial patterns exhibiting segregation. Matano and Mimura [103] showed that,if the diffusion coefficients d1 and d2 are sufficiently large, bounded station-ary solutions must be constant. A segregation result in a triangular diffusionsystem (i.e. α21 = 0) is shown by Mimura [108].

An important paper on the interplay between diffusion and cross-diffusionwas published by Lou and Ni [97], and in the following, we will describe theirresults (also see the review [115]). We consider the weak competition caseB > A > C (see Section 3), since in this case, the system (7)-(9) withoutcross-diffusion has no nonconstant steady states. It holds:

• Let B > A > (B + C)/2 and d2 belonging to a proper range. Then, ifα21 ≥ 0 is fixed and α12 is sufficiently large, there exists a nonconstantsteady state.


• Let B > A > C. If one of the diffusion constants d1 or d2 is sufficiently large(compared to the cross-diffusion coefficients α12 and α21), the constantvector s∗, defined in (3), is the only positive steady state.

This means that there are nonconstant steady states if cross-diffusion is suf-ficiently large, and large diffusion coefficients tend to eliminate any pattern.In the strong competition case B < A < C, the situation is more complicatebut cross-diffusion has similar effects in helping to create pattern formation;see [97] for details.

The existence of positive steady states for coefficients (a1, a2) lying in acertain region was shown by Ruan [127], generalizing results by Mimura [108]and Li and Logan [90]. See also [25] for the same issue. Hopf bifurcations ofcoexistence steady states have been analyzed by Kuto [83]. The existence andnonexistence of coexistence steady states of the mutualistic model was ana-lyzed by Pao [118], later generalized by Delgado et al. [32]. In recent years,several works were considered with three-species cross-diffusion systems pro-viding sufficient conditions for the existence of nonconstant positive steadystates, see [24, 52, 95, 128, 129].

Partial existence theory. First, we report the mathematical difficulties inthe analysis of the time-dependent system (7)-(9). Its diffusion matrix

A =

(

d1 + 2α11u + α12v α12uα21v d2 + 2α22v + α21u

)

(10)

is neither symmetric nor in general positive definite such that even the localexistence of solutions is far from being trival. Moreover, there exists generallyno maximum principle for parabolic systems, which would allow one to derivebounds on the solutions. Finally, it is not clear how to prove the nonnegativityor positivity of the population densities, which is desirable from a biologicalpoint of view. It is therefore not surprising that the first existence resultsin the literature were concerned with special cases, and partial results wereobtained only: local-in-time existence, and global-in-time existence for smallcross-diffusion constants or for triangular diffusion matrices (α21 = 0).

One of the first results on the existence of transient solutions was achievedby Kim [74]. He proved the local existence of nonnegative solutions to theone-dimensional cross-diffusion system without self-diffusion. If all diffusioncoefficients are set equal to one, he obtained the global existence of solutions.The reason for the last result is easy to see: Taking the difference of theequations

ut − ∆(u + uv) = u(a1 − b1u − c1v),

vt − ∆(v + uv) = v(a2 − b2u − c2v),

the difference w := u − v solves

wt − ∆w = u(a1 − b1u − c1v) − v(a2 − b2u − c2v).

10 Ansgar Jungel

Thus, for given u and v, the function w is a solution to the linear heat equa-tion and can be easily controlled by the right-hand side. Kim derived H2(Ω)estimates for u, v, and w, which enabled him to extend the local solution forany time. Another result is due to Deuring [34]. For sufficiently small cross-diffusion parameters α12 and α21 (or equivalently, sufficiently small initialdata) and vanishing self-diffusion coefficients α11 = α22 = 0, he proved theglobal existence of solutions to (7)-(9).

Several papers are concerned with the global existence of solutions in thespecial case α21 = 0. Then the diffusion matrix is triangular and the equa-tion (8) for the second species is only weakly coupled through the reactionterms, considerably facilitating the analysis. We mention some works in thisdirection: Pozio and Tesei [123] have assumed rather restrictive conditions onthe reaction terms for their global existence results. The conditions have beenweakened later by Yamada [153]. Redlinger [125] has neglected self-diffusionbut he has chosen general reaction terms of the form uf(u, v) and vg(u, v);Yang [154, 155] generalized his results. Lou, Ni, and Wu [98] examined thecase of one and two space dimensions and included self-diffusion in the equa-tion for v. The system in any space dimension was treated by Choi, Lui, andYamada [26] under the hypotheses that the cross-diffusion in the equation foru is sufficiently small and that there is no self-diffusion in the equation forv. This work was generalized by Van Tuoc [140], assuming that the cross-diffusion parameter of one species is smaller than the self-diffusion coefficientof the other species.

Considerable progress was made by Amann [6]. He derived sufficient condi-tions for the solutions to general quasilinear parabolic systems to exist globallyin time. The question if a given (local) solution exists globally is reduced tothe problem of finding a priori estimates in the space W k,p(Ω). More precisely,if the local solution is bounded in W 1,p(Ω) uniformly in (0, T ), where T > 0is the maximal time of existence and p > n (n being the space dimension), orif one can control the L∞ and Holder norms, then the solution exists globally.These results have been applied to triangular cross-diffusion systems, see theworks by Amann [6] and later by Le [84].

Full diffusion matrices were considered in [92, 147, 151]. Li and Zhao [92]proved the global existence of solutions under some restrictions on the (cross)diffusion coefficients, whereas Wen and Fu [147] achieved related results forsystems with m species. Yagi [151] studied the two-dimensional problem with-out self-diffusion and showed a global existence result under the conditions

0 < α12 < 8α11, 0 < α21 < 8α22, α12 = α21.

This hypothesis is easily understood by observing that in this case, the diffu-sion matrix A in (10) is positive definite,

z⊤Az ≥ mind1, d2|z|2 for all z ∈ R2.

If the above condition does not hold, there are choices of the parameters suchthat A is not positive definite.


Finally, we mention the work [47] by Fu, Gao, and Cui, who have provedthe global existence of classical solutions to a three-species cross-diffusionmodel with two competitors and one mutualist.

Global existence theory. Remarkably, the positive definiteness of A is notnecessary to obtain global existence of solutions to (7)-(9). The first globalexistence result for the one-dimensional cross-diffusion system without anyrestriction on the diffusion coefficients (except positivity) was achieved byGaliano et al. [49]. Their result is based on two observations which are de-scribed in the following.

First, there exists a transformation of variables which symmetrizes theproblem. This transformation reads as

u = ew1/α21, v = ew2/α12.

Then system (7)-(8) transforms into

∂

∂t

(

ew1

ew2

)

− div(B(w)∇w) = f(w),

where

w =

(

w1

w2

)

, f(w) =

(

ew1(a1 − b1ew1/α21 − c1e

w2/α12)ew2(a2 − b2e

w1/α21 − c2ew2/α12)

)

.

The new diffusion matrix

B(w) =

(

(d1 + 2α11α−121 ew1 + ew2)ew1 ew1+w2

ew1+w2 (d2 + 2α22α−112 ew2 + ew1)ew2

)

,

is symmetric and positive definite,

det B(w) ≥ d1ew1 + d2e

w2 ,

i.e., the operator div(B(w)∇w) is elliptic for all positive di and nonnegativeαij (but not uniformly in w). Another advantage of the above transformationis that if L∞ bounds for wi are available, the functions ui are automati-cally positive. This idea circumvents the maximum principle which cannotbe applied to the present problem. We remark that exponential changes ofunknowns have been used in other models to prove the existence of nonnega-tive or positive solutions to elliptic or parabolic systems and to higher-orderequations, see [50, 61, 68, 69].

The second idea is that the cross-diffusion system admits a priori estimatesvia the functional

E1(t) =

∫

Ω

( u

α12(log u − 1) +

v

α21(log v − 1)

)

dx. (11)

Due to the similarity to the physical entropy, we call this functional an entropy.It satisfies the so-called entropy inequality

12 Ansgar Jungel

dE1

dt+ 2

∫

Ω

(2d1

α12|∇

√u|2 +

α11

α12|∇u|2 +

2d2

α21|∇

√v|2 +

α22

α21|∇v|2

+ 2|∇√

uv|2)

dx ≤ C1, (12)

where C1 > 0 is a constant depending only on the reaction parameters. Thisprovides, for positive self-diffusion parameters, H1(Ω) estimates for u and v.

It is not a coincidence that the symmetrizable system (7)-(8) possesses anentropy functional; see below for a discussion of the relation between symme-try and entropy.

Clearly, the above computation can be made rigorous only if u and v arenonnegative (or even positive) functions. For this, we need to show L∞ boundsfor wi, which cannot be deduced from the above entropy estimate. The ideaof [49] was to employ another “entropy” functional,

E(t) = E1(t) + γE2(t), E2(t) =

∫

Ω

( 1

α12(u − log u) +

1

α21(v − log v)

)

dx,

where γ = 4mind1/α12, d2/α21. Indeed, employing Young’s inequality, wearrive to

dE2

dt≤ −

∫

Ω

( d1

α12|∇ log u|2 +

d2

α21|∇ log v|2 + 8

α11

α12|∇

√u|2

+ 8α22

α21|∇

√v|2

)

dx +

∫

Ω

(

|∇√

u|2 + |∇√

v|2)

dx

+

∫

Ω

(−a1 − a2 + (b1 + b2)u + (c1 + c2)v)dx.

The second integral can be estimated by the corresponding terms in E1, andthe last integral is controlled by the reactions terms coming from dE1/dt.After some manipulations we arrive to

dE

dt+

∫

Ω

(γd1

α12|∇ log u|2 +

γd2

α21|∇ log v|2

)

dx ≤ C2,

where C2 > 0 depends again only on the reaction constants. This estimategives a bound for log u and log v in L2(0, T ;H1(Ω)). Up to now, the argumentsare valid in any space dimension n. Now, we need the assumption n = 1.Indeed, in this case, the space H1(Ω) embeddes continously into L∞(Ω), thusshowing that u = ew1 and v = ew2 are positive.

Unfortunately, there are no L∞ bounds for wi in time. Therefore, Galianoet al. [49] have discretized the cross-diffusion system in time (by the backwardEuler scheme), obtaining a sequence of elliptic equations, which are solvedrecursively in time. Since time is discrete, the semidiscrete population densitiesare strictly positive. The above a priori estimates are sufficient to pass to thelimit τ → 0 of the time discretization parameter τ , using Aubin compactness

results for the sequence of semi-discrete solutions exp(w(τ)1 ) and exp(w

(τ)2 ).


Since the compactness holds for exp(w(τ)i ) and not for w

(τ)i , we loose the

boundedness of log u and log v and thus the strict positivity of u and v inthe limit τ → 0, but still obtaining the nonnegativity of u and v as limits ofsequences of positive functions.

The assumption of one space dimension to define the exponentials ewi

is crucial in the above argument. In order to deal with multi-dimensionalproblems, Chen et al. [21] have used another idea. They have discretized thecross-diffusion term ∆(uv) = div(uv log(uv)) by the finite differences

D−h(

χhuvDh(log(uv)))

,

with D−h being an approximation of the divergence, Dh an approximation ofthe gradient, and χh the characteristic function of x ∈ Ω : dist(x, ∂Ω) > h.This discretization is inspired from [77], in which a cross-diffusion problemfrom semiconductor theory was studied. The approximate problem possessesan entropy inequality similar to (12) but with the term |∇√

uv|2 replaced byχhuv|Dh log(uv)|2. In order to avoid problems arising from the logarithm, uand v are replaced by u+ + η and v+ + η, respectively, where u+ = max0, uis the positive part of u and η > 0 is a parameter. The nonnegativity ofthe aproximate solutions is proved by taking the negative part (u−, v−) =(min0, u,min0, v) as a test function in the weak formulation of the system,yielding an estimate of the type

‖u−(·, t)‖L2(Ω) + ‖v−(·, t)‖L2(Ω) ≤C

| log η| uniformly in t > 0.

In the limit η → 0 this gives u− = v− = 0 in Ω × (0,∞) and hence the non-negativity of the population densities. Further approximations are necessary:The system is discretized in time by the backward Euler scheme, and thediffusion coefficients in B(w) are approximated by bounded functions. Thenthe discrete entropy estimates allow for the limit of vanishing approximationparameters.

In [21], the self-diffusion coefficients need to be positive in order to deduceH1(Ω) bounds for u and v. This condition has been weakened later in [22]by allowing for vanishing self-diffusion, α11 = α22 = 0. Then there are noH1(Ω) bounds for u (and v) which are needed for the Aubin compactnessargument. This problem is solved by exploiting the bounds on

√u. Indeed,

by the Gagliardo-Nirenberg inequality, u is bounded in L4/3(0, T ;W 1,4/3(Ω)).Together with an L1(0, T ; (Hs(Ω))∗) bound for ut, where (Hs(Ω))∗ denotesthe dual space of Hs(Ω), one can apply the Aubin lemma [137]. However,there remains a problem: The L1(0, T ; (Hs(Ω))∗) bound for ut does not implyweak compactness in the context of Lp spaces since L1 is not reflexive. Thisproblem is overcome by using a weak compactness result in L1 due to Yoshida(see Lemma 6 in [22]).

We mention that the approximation procedure in [22], compared to [21],has been simplified. Indeed, instead of discretizing the cross-diffusion terms,

14 Ansgar Jungel

a Galerkin approximation, together with a semi-discretization in time, is per-formed. In order to deal with a possible degeneracy of the diffusion matrix, theelliptic regularizations ε∆w1 and ε∆w2 are added. Thus, there are three in-stead of four approximation levels needed in [21]: the dimension of the Galerkinspace, the time-discrete parameter, and the regularization parameter ε > 0.

Another (simpler) regularization was suggested by Barrett and Blowey[10]. They have derived entropy-type estimates by using an approximate en-tropy functional Eε, which is quadratic for very small and very large popula-tion densities, together with a truncation of the diffusion coefficients to ensureuniform ellipticity. This approximation gives

‖u−(·, t)‖L2(Ω) + ‖v−(·, t)‖L2(Ω) ≤ C√

ε,

and hence u, v ≥ 0 in the limit ε → 0.The above procedures fail in the whole-space case Ω = R

n. Indeed, it isnatural to assume that the solutions (u, v) decay to zero as |x| → ∞ whichimplies that log u(x, t) = ∞ and log v(x, t) = ∞ at infinity. But then thepartial integrations needed to derive the entropy estimates have to be justified.This difficulty was overcome by Dreher [36] by introducing a modified entropywhich compares the solution (u, v) against an exponentially decaying weightfunction. Dreher also used a semi-discretization in time but a higher-orderelliptic regularization with the operator ∆4.

Relation between symmetry and entropy. Above, we have shown thatthe cross-diffusion system (7)-(8) can be “symmetrized”, by a change of vari-ables, and that it possesses an entropy functional. This is not a coincidence.In fact, it is well known from the theory of hyperbolic conservation laws thatthe existence of a symmetric formulation is equivalent to the existence of anentropy functional [72]. This equivalence was reconsidered for parabolic sys-tems by Degond et al. [30] and exploited for the mathematical analysis ofenergy-transport systems in nonequilibrium thermodynamics [29].

Consider the system

ut − div(A(u)∇u) = f(x, u) in Ω, t > 0, u(·, 0) = u0 ≥ 0 in Ω, (13)

supplemented with homogeneous Neumann boundary conditions. The sameresults hold for Dirichlet and mixed Dirichlet-Neumann boundary conditions[29]. The vector div(A(u)∇u) is defined by its components

∑

j div(Aij(u)∇uj).

The diffusion matrix A(u) ∈ Rm×m may be neither symmetric nor positive

definite. Systems (13) have been studied by Alt and Luckhaus [5] but onlyfor positive definite matrices A(u) and for solutions u which may change sign.The case of indefinite diffusion matrices can be mathematically treated if thereexists a change of unknowns u = b(w) with b : R

m → Rm such that

B(w) := A(b(w))b′(w) is symmetric and positive definite.

To ensure that (13) is of parabolic type, we assume further that the functionb is monotone and a gradient, i.e., (b(w1) − b(w2)) · (w1 − w2) ≥ 0 for all w1,


w2 ∈ Rm and there exists a function χ : R

m → R such that χ′ = b. Then (13)can be reformulated as

(b(w))t − div(B(w)∇w) = f(x, b(w)).

We claim that this system admits some a priori estimates if the reaction termf can be controlled. Define the entropy

E(t) =

∫

Ω

(b(w) · w − χ(w))dx.

Then, after a formal computation,

dE

dt+

∫

Ω

(∇w)⊤B(w)(∇w)dx =

∫

Ω

f(x, b(w)) · wdx.

Since B(w) is positive definite, by assumption, the integral on the left-handside is nonnegative. If the right-hand side can be controlled, this equationprovides an a priori estimate for w. When −f is monotone in the sense off(x, b(w)) · w ≤ 0, E is even a Lyapunov functional.

In the population cross-diffusion system (7)-(8), we have b(w) = (ew1 , ew2)which is a gradient since χ(w) = ew1 +ew2 satisfies χ′ = b. (Here, we assumedthat α12 = α21 = 1, which can be achieved by a rescaling [49].) The entropybecomes

E =

∫

Ω

(

ew1(w1 − 1) + ew2(w2 − 1))

dx

=

∫

Ω

(

u1(log u1 − 1) + u2(log u2 − 1))

dx,

which is of the form (11). The advantage of the special transformation u =b(w) = (ew1 , ew2) is that it gives automatically nonnegative or even positivesolutions u. These ideas have been employed in the analysis of systems fromvarious applications, such as thermodynamics [29, 31], semiconductor theory[23], and granular materials [50].

Regularity theory and long-time behavior of solutions. Since Holdercontinuity of bounded weak solutions plays an important role in showingthe global existence of solutions in the framework of Amann, several authorsproved the Holder regularity of solutions under suitable assumptions. For thetriangular system, Le [84] proved that if the L∞ norm of v and the Ln normof u (n being the space dimension) can be controlled in the sense of [84], thentheir Holder norms are also controlled. Furthermore, if the control is possiblefor every solution, there exists a global attractor with finite Hausdorff dimen-sion. The Holder continuity results have been generalized by Le in [85, 86] toinclude more general diffusion coefficients.

Shim derived uniform L∞ bounds for the solutions under additional condi-tions on the diffusion constants in the one-dimensional setting, and he showed

16 Ansgar Jungel

the convergence to the steady states as t → ∞ [133, 134, 135, 136]. Theexistence of an exponential attractor (i.e., a compact, finite-dimensional, pos-itively invariant set which attracts any bounded set at an exponential rate)was shown by Yagi [152]. Kuiper and Dung [82] proved the existence of aglobal attractor for cross-diffusion systems with general diffusion functions.For further results in this direction, we refer to the works [87, 88] of Le andcoworkers.

The long-time behavior of solutions is connected with the existence ofconstant or nonconstant steady states, as reviewed above. Lou and Ni [97]discussed the question if nonconstant steady states in the weak competitioncase still exist if both cross-diffusion constants are strong but qualitativelysimilar. (The answer is yes if only one of the cross-diffusion parameters issufficiently large.) A partial answer is given by Chen et al. [22] by studyingthe long-time behavior of the solutions. More precisely, they showed that forvanishing intra-specific competitions b2 = c1 = 0, which is a special case ofweak competition, only constant steady states exist no matter how strong thecross-diffusion coefficients are.

The argument is as follows. Define the relative entropy

ER(t) =

∫

Ω

( u

α12φ( u

u∗

)

+v

α21φ( v

v∗

))

dx,

where φ(s) = s(log s − 1) + 1 for s ≥ 0 and (u∗, v∗) = (a1/b1, a2/c2) arehomogeneous steady states. Then a computation shows that for all t ≥ s > 0,since b2 = c1 = 0,

dER

dt+ C

∫

Ω

(

|∇√

u|2 + |∇√

u|2)

dx

≤ −∫

Ω

(

b1u(u − u∗)(log u − log u∗) + c2v(v − v∗)(log v − log v∗))

dx ≤ 0,

where C > 0 depends on the diffusion coefficients. Now, if (u, v) is a stationarysolution to the cross-diffusion system, clearly dER/dt = 0 and

‖∇√

u‖2L2(Ω) + ‖∇

√u‖2

L2(Ω) ≤ 0.

Thus, u and v are constant in Ω. Since they satisfy the stationary equations,we conclude that u(a1 − b1u) = v(a2 − c2v) = 0 in Ω. Hence, either u = 0 oru = a1/b1 and either v = 0 or v = a2/c2. In both cases, (u, v) is a constantstationary solution.

Numerical approximation. There are only few papers concerned withthe numerical discretization of the cross-diffusion system (7)-(8). The one-dimensional stationary problem was numerically solved in [48] using semi-implicit finite differences. The numerical experiments confirm that segrega-tion of the species occurs for sufficiently large cross-diffusion parameters. Asmentioned above, a semi-discretization in time was proposed in [49], and the


convergence of the semi-discrete solutions to a continuous one was proved.Barrett and Blowey [10] presented a convergence proof for a fully discretefinite-element approximation. Very recently, Gambino et al. [51] have dis-cretized the one-dimensional problem by a particle approximation in spaceand an operator-splitting method in time together with an Alternating Direc-tion Implicit (ADI) scheme.

5 Structured population models

The population models in the previous sections are based on the assumptionthat all individuals of a certain species are identical. However, populationstypically consist of individuals which can be distinguished by various variablessuch as age, size, gender, etc. In this section we review some models whichinclude an age or size structure, following [119, 121]. Other structured modelscan be found in [28, 100, 106].

Age-structured models. A model in which the vital rates depend on theage variable was first given by Sharpe and Lotka [130], known as the Lotka-McKendrick or McKendrick-von Foerster equation [46, 105]. Let u(a, t) be theage density of a single-species population, where a ≥ 0 is the age and t ≥ 0the time. Denote by b(a) and µ(a) the birth and death rate, respectively, ofthe species of age a. Then the change du = utdt + uada of the population ofage a in a small increment of time dt equals −µ(a)udt [113, Sec. 1.7]. Here,ut = ∂u/∂t and ua = ∂u/∂a. The birth rate b(a) only contributes to u(0, t)since species are born at age a = 0. Dividing the equation for du by dt andnoting that da/dt = 1 since a is the chronological age, u(a, t) satisfies thefollowing hyperbolic equation with a nonlocal boundary condition:

ut + ua + µ(a)u = 0, t > 0, u(a, 0) = u0(a), a ≥ 0, (14)

u(0, t) =

∫ ∞

0

b(a)u(a, t)da, t > 0. (15)

This equation is sometimes referred to as the renewal equation since it de-scribes how a population is renewed [59]. When we assume that the life spanis finite, a ∈ [0, a+] with a+ < ∞, this problem can be formulated as aVolterra equation of second kind (for the variable B(t) := u(0, t)), which iscalled the renewal or Lotka equation [64]. Using this formulation, it can beshown [64, 146] that the solution of the renewal equation has the asymp-totic behavior B(t) = B0 exp(λt)(1 + o(t)), where B0 ≥ 0, λ ∈ R, and o(t)tends to zero as t → ∞. This means that the number of newborns changesexponentially with rate λ, at least for large time.

Mischler et al. [110] have proved the existence and long-time behavior ofsolutions to (14)-(15) without using a maximal life span condition (also see[121]). Their idea is to use a generalized relative entropy method. To illustratethis idea, we first observe that the death rate term can be eliminated via the

18 Ansgar Jungel

transformation w(a, t) = em(a)u(a, t), where m(a) =∫ a

0µ(a)da, since w solves

the equationwt + wa = 0, a ≥ 0, t > 0.

Therefore, we may assume, without any loss of generality, that µ(a) = 0. It isconvenient to introduce the variable v(a, t) = e−λtu(a, t), where (U, V, λ) withU > 0, V ≥ 0, and λ > 0 are the first eigenelements of

Ua + λU = 0, a ≥ 0, U(0) =

∫ ∞

0

B(a)U(a)da,

∫ ∞

0

U(a)da = 1,

− Va + λV = V (0)B(a), a ≥ 0,

∫ ∞

0

U(a)V (a)da = 1.

The function U is the eigenfunction associated with the operator in (15),with the first eigenvalue λ, and V is the eigenvector of the same eigenvalueassociated with the adjoint operator. The factor e−λt scales the populationdensity in such a way that v stays bounded for all time. In other words,the population density grows exponentially with rate λ > 0, also called theMalthus parameter.

The following entropy inequality holds for all convex functions H satisfyingH(0) = 0:

∫ ∞

0

U(a)H(v(a, t)

U(a)

)

V (a)da ≤∫ ∞

0

U(a)H(u0(a)

U(a)

)

V (a)da, t ≥ 0.

This property allows one to show the long-time limit of v(·, t). The limit isexpected to be proportional to the steady state U . In fact, if u0 is boundedby U , up to a factor, it follows that [121, Sec. 3.6]

limt→∞

∫ ∞

0

|v(a, t) − v0U(a)|V (a)da = 0,

where v0 =∫ ∞

0u0V da. Exponential decay can be shown under a (restrictive)

lower bound on the birth rate B [121, Sec. 3.7], and the decay rate dependson this lower bound.

To some extend, the Lotka-McKendrick model is an age-structured versionof the Malthus model, introduced in Section 2. The drawback of both modelsis the unlimited exponential growth of the population. In the literature, manyextensions and variants of the Lotka-McKendrick model have been proposed.Here, we mention some of them.

A simple model for a cell division cycle with a single phase is the followingvariant of the Lotka-McKendrick system [121, Sec. 3.9]:

ut + ua + µ(a)u = 0, t > 0, u(0, t) = 2

∫ ∞

0

µ(a)u(a, t)da,

with the initial condition u(·, 0) = u0, where µ is the mitosis (cell division)rate. In this situation, a cell is withdrawn from the differential equation at age


a with the rate µ(a) and it creates two daughter cells at age a = 0 with thesame rate. The (mathematical) advantage of this model is that its solutionsdecay exponentially fast to the steady state under the natural assumptionthat very young cells do not undergo mitosis (thus avoiding the restrictiveassumption on B needed in the model (14)-(15)).

When the birth and death rates depend on certain variables (sizes)s1(t), . . . , sm(t), which represent different ways of weighting the age distri-bution, we arrive at the system [119]

ut + ua + µ(a, s1, . . . , sm)u = 0, a ≥ 0,

u(0, t) =

∫ ∞

0

b(a, s1, . . . , sm)u(a, t)da,

si(t) =

∫ ∞

0

ci(a)u(a, t)da, i = 1, . . . ,m, t > 0,

together with an initial condition for u(·, 0). For i = 1 and c1(a) = 1, we obtainthe Gurtin-MacCamy model introduced in [59]. The existence and uniquenessof solutions to this model is proved in [146] using a semigroup approach.A numerical analysis was performed in [139]. A special case is given by thelogistic-growth model i = 1 with µ(a, s1) = µ0(a). The age-specific fertility b isassumed to be nonnegative and decreasing with b(a,∞) = 0. This means thatthe birth rate decreases when the weighted population average s1 becomeslarger.

In order to model the spatial dispersal of population species, one mayinclude diffusive terms leading to equations of the form

ut + ua + µ(a)u = div(d∇u), x ∈ Ω, a ≥ 0, t > 0,

where d > 0 is a diffusion coefficient and (Dirichlet or Neumann) boundaryconditions for x have to be imposed on ∂Ω. The population density u de-pends on the spatial variable x, the age a, and time t. One of the first worksin this direction is due to Gurtin [58]. Gurtin and MacCamy [60] presentedage-structured models with random diffusion or directed diffusion to avoidcrowding. Mathematical results are presented, for instance, by MacCamy [99]who studies nonlinear diffusion processes yielding porous-medium-type diffu-sion equations. Di Blasio [14] proved the existence and uniqueness of solutionsto age-structured diffusion models, and Busenberg and Iannelli [18] analyzeda degenerated diffusion problem. A finite-difference scheme was proposed byLopez and Trigiante [93]. Kim and Park [75] used finite differences in thecharacteristic age-time direction and finite elements in the spatial variable.A variable time step method was chosen by Ayati [8], and Pelovska [120]developed an accelerated explicit scheme.

Size-structured models. For some organisms, age is not the most relevantparameter, but rather the cell mass or its size. This leads to size-structuredpopulation models in the size parameter x. In the following, we review some

20 Ansgar Jungel

models from [121]. We distinguish between symmetric mitosis (two daughtercells of size x emerge from a mother cell of size 2x) and asymmetric mitosis(the emerging daughter cells have different size). In the symmetric case, thepopulation number u(x, t) may evolve according to

ut + ux + b(x)u(x, t) = 4b(2x)u(2x, t), u(0, t) = 0, u(x, 0) = u0(x), (16)

where x ≥ 0, t > 0, and b is the birth rate. The factor 4 can be understoodby computing the change of the population number,

d

dt

∫ ∞

0

u(x, t)dx = 4

∫ ∞

0

b(2x)u(2x, t)dx −∫ ∞

0

b(x)u(x, t)dx

=

∫ ∞

0

b(x)u(x, t)dx,

which increases with rate b(x). Similar to the age-structured model (14)-(15),the mathematical analysis relies on a certain eigenvalue problem with the firsteigenvalue λ (the Malthus parameter) and the corresponding eigenfunctionsU , V of the stationary and dual problem, respectively. Perthame and Ryzhik[122] proved that for constant birth rate b(x) = b0, it holds λ = b0, V = 1,and

∥

∥

∥e−b0tu(·, t) − U

∫ ∞

0

u0dx∥

∥

∥

L1(0,∞)≤ Ce−b0t, t ≥ 0,

and the constant C > 0 depends on the initial datum u0. A similar resultholds for nonconstant birth rates, see [122]. For a numerical solution, we referto [35].

When the mitosis is asymmetric, we have to change the term on the right-hand side of (16):

ut + ux + b(x)u(x, t) =

∫ ∞

x

β(x, y)u(y, t)dy.

This models the division of a mother cell of size y into two daughter cells ofsizes x and x − y with rate β(x, y). Under suitable assumptions on b, β, andu0, the exponential decay of (a rescaled version of) u(x, t) to the steady stateis proved by Michel et al. [107]. Furthermore, an equation in which the effectsof cell division and aggregation are incorporated by coupling the coagulation-fragmentation equation with the Lotka-McKendrick model was analyzed byBanasiak and Lamb [9].

In the literature, also size-structured models with (v(u)u)x instead of ux

in (16) and different expressions on the right-hand side have been employed,interpreting v(u) as a growth rate, for instance in [71] for optimal harvestmodeling and in [42, 43] for linear stability and instability results of stationarysolutions. For more models and references, we refer to the monographs of Metzand Dieckmann [106] and Cushing [28].


6 Time-delayed population models

The change in the population number of a species may not respond immedi-ately to changes in its population or that of an interacting species, but ratherafter a certain time lag. Time delay in population dynamics models, for in-stance, the gestation or maturation time of a species or the time taken forfood resources to regenerate. Hutchinson [63] postulated the equation

du

dt= u(t)

(

a − bu(t − T ))

, t > 0, u(0) = u0, (17)

where a, b > 0, which was analyzed by May in [104]. Whithout delay, T = 0,we recover the logistic-growth equation with the stable steady state u = a/b.In case of delay, T > 0, May discovered an interplay between the stabilizingresource limitation and the destabilizing time lag. More precisely, if aT < π/2,u = a/b is still a stable steady state, which becomes unstable if aT > π/2.When the time lag is not constant, one may employ the distributed delayequation

du

dt= u(t)

(

a −∫ t

−∞

b(t − s)u(s)ds)

, t > 0, u(0) = u0.

The model of May was generalized and applied to the modeling of Australiansheep-blowfly populations by Gurney et al. [57]. Diffusive versions can befound in [138, 148]. Time delay may be used to model immature and maturestages; see [2] for an example. For the analysis of a system of delayed equations,we refer to [4]. More references can be found in the monograph of Kuang [81].

Spatial structures have been also considered in delayed models. A simplediffusive extension of the Hutchinson equation (17) is given by

ut − ∆u = u(t)(

a − bu(t − T ))

.

More elaborate models have been proposed by Gourley and Kuang [54]. Gour-ley et al. [56] argue that diffusion and time delays are not independent of eachother, since individuals may be at different points in space at past times.Britton [16] has suggested a delay term which involves a weighted spatialaveraging over the (infinite) domain in order to account for the drift of the in-dividuals from all possible positions at previous times to the present position.The equation becomes

ut − ∆u = u(

1 + αu − (1 + α)g ∗ u)

, x ∈ Rn, (18)

where g is a given function and g ∗ u is a convolution in the spatio-temporalvariables. The term αu represents the advantageous local aggregation dueto high mating probability, for instance; the convolution −(1 + α)g ∗ u withα > −1 models the intra-specific competition due to resource limitations ina neighborhood of the original position. When g is a delta distribution, we

22 Ansgar Jungel

recover the Fisher equation (see Section 3). The particular choice g(x) =e−|y| is the Green’s function for an ordinary differential equation, and (18)can be reduced to a system of two local equations analyzed by Billingham[13]. General kernels were considered by Deng [33], establishing the existence,uniqueness, and long-time behavior of solutions. Furthermore, Gourley andBritton [53] studied the linear stability of a related predator-prey system.Equation (18) is an example of a parabolic equation with a functional term;general nonlocal parabolic problems were treated by Redlinger [124].

Population models in bounded domains have been proposed by Gourleyand So [55]. The nonlinear stability of traveling wavefronts in a related single-species model was proved by Li, Mei, and Wong [91]. Xu and Zhao [150] showedthe existence of a global attractor of a nonlocal reaction-diffusion model withtime delay. A survey on nonlocal population models, induced by time delays,and more references can be found in [56].

An equation with a time lag in the spatial variable has been proposedrecently by Berestycki et al. [11] in order to study the impact of climatechange on the dynamics of an affected species:

ut − ∆u = uf(x − cte, u), x ∈ Rn, t > 0,

where c is a constant and e is a unit vector. The space dependence of thegrowth rate f is affected by the time under the action x − cte, i.e., the zoneswith favorable climate change shift in the direction e with speed c. Heuris-tically, we expect that populations manage to persist by migrating in thedirection e. Indeed, it is shown in [12] that traveling wave solutions of thetype u(x, t) = v(x − cte) exist if the climate shift is not too large (i.e., c > 0is sufficiently small), otherwise there is extinction.

References

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